10 



years is based on direct numerical solutions of the 

 complete partial differential equations that 

 describe the flow phenomena arising in the transi- 

 tion process. This approach became feasible with 

 the rapid progress in the development of large, 

 high-speed digital computers . 



The main difficulties here arise from the fact 

 that these flow phenomena can be adequately 

 represented only when the complete Navier-Stokes 

 equations (or certain modifications thereof) are 

 used. Thus, this approach requires the solution 

 of the Navier-Stokes equations for strongly time- 

 varying flow fields, due to the highly unsteady 

 nature of the transition processes. Additionally, 

 complications increase because the numerical 

 solutions have to yield reliable results for 

 relatively high Reynolds numbers (higher than the 

 critical Reynolds number) to allow onset of 

 transition. For a numerical solution procedure it 

 is therefore necessary to allow for adequate 

 resolution of the large temporal and spatial 

 gradients resulting from the occurrence of thin 

 time-varying fluid layers with large gradients 

 close to solid walls. 



The development of finite-difference methods, 

 which are applicable for such complex, unsteady 

 flow phenomena as those occuring in laminar 

 turbulent transition, is associated with numerous 

 difficulties which will be elaborated upon in this 

 paper. Because of these difficulties relatively 

 few previous attempts based on such an approach be- 

 came known. Reasonably successful earlier investiga- 

 tions of this kind (based also on finite-difference 

 solutions) are reported, for example, for incom- 

 pressible flows in a boundary layer [De Santo 

 and Keller, (1962)], for Poiseuille and plane 

 Poiseuille flow [Dixon and Heliums (1957) , Crowder 

 and Dalton (1971)] and for a compressible boundary- 

 layer flow [Nagel (1967) ] . These earlier attempts 

 clearly demonstrated the usefulness and potential 

 of such investigations. However, due either to 

 insufficent resolution of the resulting gradients 

 and/or assumptions made concerning the basic or 

 disturbance flows, or to shortcomings of the differ- 

 ence methods used, the results of these calculations 

 were more of a qualitative nature. Therefore, 

 relatively little information could be gained 

 concerning the various phenomena arising in the 

 laminar-turbulent transition process. 



Some years ago, a research effort was initiated 

 at the University of Stuttgart aiming at the devel- 

 opment of numerical methods for the solution of the 

 Navier-Stokes equations which would be applicable 

 for detailed investigations of various aspects of 

 stability and of phenomena occurring in transition. 

 To date, an effective implicit finite-difference 

 method has evolved for the calculation of unsteady, 

 two-dimensional incompressible flows. The ap- 

 plicability of the numerical method to investigate 

 stability and two-dimensional transition phenomena 

 has been demonstrated by realistic simulations of 

 Tollmien-Schlichting waves. Detailed results of 

 these calculations are discussed elsewhere [Fasel 

 (1976) ] . With calculations involving large ampli- 

 tude disturbances [Fasel et al . (1977)] it was addi- 

 tionally shown that numerical simulations using 

 the implicit difference method yield results which 

 enable insight into certain nonlinear mechanisms 

 of the transition process. 



In this paper the major aspects of the numerical 

 approach using finite-difference methods will be 



reviewed and the present state of the developments 

 discussed. Emphasis will be placed on the advan- 

 tages of the numerical approach in general and on 

 directional options chosen for the present method. 

 Special attention will also be focused on the 

 difficulties and limitations of such simulations. 



2. SELECTION OF THE INTEGRATION DOMAIN 



For a numerical solution of the Navier-Stokes equa- 

 tions using finite-difference techniques a finite 

 domain in which the equations are being solved has 

 to be specified. The selection of the integration 

 domain determines the nature of a physical flow 

 problem to be simulated, because the boundary con- 

 ditions required along the boundaries of this domain 

 determine to a large degree the solution within the 

 domain. For reasons of simplicity, in the present 

 studies only rectangular domains of the x,y plane 

 were considered as depicted schematically in Figures 

 1 and 2 with the direction of the basic, undisturbed 

 flow being in the x-direction. Rectangular domains 

 allow relatively easy application of difference 

 methods by using simple rectangular meshes. For 

 example the rectangular domain may be a section of 

 a boundary-layer flow on a semi-infinite flat plate 

 (Figure 1) or a section of a flow between two paral- 

 lel plates (Figure 2) . 



In selecting the integration domain one has to 

 consider that boundary conditions must be found for 

 the 'artificial' boundaries B-C in Figures 1 and 2 

 and additionally for C-D in Figure 1. These con- 

 ditions should allow physically meaningful solutions 

 in the finite domain, i.e. solutions that would be 

 obtained if the domain were not made finite by 

 means of these artificial boundaries. Due to the 

 spatially elliptic (in x,y) character of the Navier- 

 Stokes equations application of finite-difference 

 methods requires boundary conditions on all bound- 

 aries of the x,y domain. Of course, in a mathemat- 

 ical sense the equations are parabolic because of 

 the time derivative (See section 3) . Selection of 

 boundary conditions for boundaries representing 

 solid walls (such as A-B in Figures 1 and 2 and C-D 

 in Figure 2) generally creates no additional diffi- 

 culty although consistent implementation in the 

 numerical scheme is frequently difficult to achieve. 

 Also, free stream boundaries such as C-D in Figure 

 1 for the boundary-layer flow can be handled in 

 satisfactory fashion (see Section 4) . 



However, the upstream (A-D) and to a larger ex- 

 tent the downstream (B-C) boundary require special 

 considerations because the specific treatment of 

 these boundaries determines the approach to be 

 taken in a prospective stability and transition 

 simulation. In selecting the boundary conditions 



' < ^^^y/'g/x'// Av/y^/yyyy//y/////^7/!?77r 



B t. 



FIGURE 1. Integration domain for boundary 

 layer on flat plate . 



